# Group (mathematics)

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A **group** is a mathematical structure consisting of set of elements combined with a binary operator which satisfies four conditions:

**Closure**: applying the binary operator to any two elements of the group produces a result which itself belongs to the group**Associativity**: where , and are any element of the group**Existence of Identity**: there must exist an identity element such that ; that is, applying the binary operator to some element and the identity element leaves unchanged**Existence of Inverse**: for each element , there must exist an inverse such that

A group with commutative binary operator is known as Abelian.

Example: the Klein Four Group consists of the set of numbers {1, -1, *i*,*-i*} under the binary operation of multiplication, where *i* is the square root of -1, the basis of the imaginary numbers.

Groups are the appropriate mathematical structures for any application involving symmetry.